#1296412

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(7,9)(11,12)(14,17)(15,19)(21,22,26,23)(24,29,32,27)(25,28,34,30)(31,36,33,35);(5,6,10,8)(7,11,9,12)(14,17)(18,20)(22,23)(27,29)(28,30)(35,36);(5,7,10,9)(6,11,8,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35);(1,2,3,4)(5,8,10,6)(7,12,9,11)(21,24)(22,27)(23,29)(25,33)(26,32)(28,36)(30,35)(31,34);(5,9,10,7)(6,12,8,11)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=gh,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1203608);